1. Field of the Invention
The present invention is directed to determining an effect of a factor on a biological system by determining a temporal change in probability of a component occupying a particular one of up to four distinguishable configurations, called states. In a particular embodiment, an effect of a factor (such as presence of a ligand or introduction of a voltage change) on an ion channel in a membrane of a cell is determined by determining a temporal change in probability that the ion channel is in one of four distinguishable states.
2. Description of the Related Art
Many biological systems important to a normal or diseased processes in a cell can be described as dependent on the population of different states of each of one or more different components, especially protein molecules. A protein molecule typically has many possible configurations, both absent external forcing and in response to the presence of external factors that exert some force on the protein. External factors can take many forms, from the application of an external voltage to the presence of ligands that bind at one or more sites on the protein. The probability that a protein is in one configuration or another is affected by these factors. The probability can be expressed by repeating an experiment many times with the same protein starting configuration and the introduction of the same factor and counting the occurrence of each state in the group of experiments. By the same token, the probability can be expressed by determining the percentage of a large number of the same protein that occupies each of the states. The bulk process of the cell is then dependent in part on the percentage of the population in each of the states.
For example, a protein that resides in a cell interior is acted upon by a therapeutic pharmacological agent when the protein is in one configuration, but not in a different configuration. The effectiveness of the therapeutic agent then depends on the probability that the protein is in a particular state, which corresponds to the population of the different states for a large number of molecules of that protein.
In one well known and heavily studied example, the protein molecule serves as a gated ion channel in a cell membrane that blocks passage of a certain ion in one or more states and allows passage of the certain ions through the membrane in one or more other states that can be induced by application of a certain voltage as an external gating factor. The controlled passage of such ions is essential in some very important bulk cell processes, such as the conduction of electrical impulses in nerve cells and muscle cells. The open state of an ion channel can be measured by a current that flows through the channel. When a channel is closed or inactive, no ions flow and current is zero. When an ion channel is inactive, it does not respond to the external gating factor by switching to an open state.
FIG. 1 is graph 100 that illustrates the measured responses of an ion channel to a voltage change. Graph 100 includes a trace 110 of voltage with time, eight individual traces 120a, 120b, 120c, 120d, 120e, 120f, 120g, 120h (collectively referenced herein as individual traces 120) of ion current with time for a single ion channel, and an average trace 130 of ion current with time for 40 consecutive single channel traces. FIG. 1 is from L. Goldman, “Stationarity of Sodium Channel Gating Kinetics in Excised Patches from Neuroblastoma N1E 115,” Biophysical Journal, vol. 69, pp 2364 to 2368, December 1995 (hereinafter Goldman I), the entire contents of which are hereby incorporated by reference as if fully set forth herein. A horizontal axis scale 102 of graph 100 indicates an amount of time associated with the scale distance along a horizontal direction for trace 110 and traces 120 in graph 100. The length of scale 102 corresponds to 4 milliseconds (ms, 1 ms=10−3 seconds). A horizontal axis scale 103 of graph 100 indicates an amount of time associated with the scale distance along a horizontal direction for trace 130. The length of scale 103 corresponds to 5 ms. The vertical change in voltage for trace 110 corresponds to a voltage step from −120 milliVolts (mV, 1 mV=10−3 Volts) to −30 mV, which is held for 45 ms. A current is measured for a limited time after the voltage change in 7 traces (trace 120a, trace 120b, trace 120d, trace 120e, trace 120f, trace 120g, trace 120h) of the eight traces 120, as indicated by current in a negative direction through the membrane at trace portions 125a, 125b, 125d, 125e, 125e, 125f, 125g, corresponding to the 7 traces, respectively. Such a current indicates the single ion channel is in an open state. The current associated with an open state in traces 120 is indicated by the vertical scale 104 that indicates a current of 3 picoAmperes (pA, 1 pA=10−12 Amperes). As can be seen from traces 120, the ion channel under study in FIG. 1 is more probably in the open state shortly after the voltage change, and then is more probably in a non-open state after about 20 ms. A more stable measurement of current is obtained by averaging over 40 such measurement to produce average trace 130. The current associated with average trace 130 is smaller than the individual measured currents, as indicated by the vertical scale 106 for trace 130, which corresponds to only 0.4 pA.
The features of average trace 130 include an exponentially increasing current in a negative direction along portion 131, and an exponentially decreasing current in the negative direction along portions 132 and 133. The response of such proteins to the presence or absence of external forcing is the object of efforts to predict and control cell processes, for both diagnosis and treatment of disease. Example uses include drug response and drug screening applications. The features of average trace 130 not only represent measured current changes but also indicate the temporal change in the probability that the ion channel is in the open state of the three or four states used to explain the membrane behavior. Viewed in this light, FIG. 1 shows an exponentially increasing probability of an open state along portion 131, and an exponentially decreasing probability of the open state along portions 132 and 133.
Observations of the type depicted in FIG. 1 have been explained by proposing three states for the protein serving as the ion channel. Other observations showing a delay in response to a voltage change have been explained using four or more states, including two or more closed or inactivation states (e.g., see R. Hahin and L. Goldman, “Initial Conditions and the Kinetics of the Sodium Conductance in Myxicola Giant Axons, I. Effects on the Time-Course of the Sodium Conductance,” Journal of General Physiology, v72, pp 863-877, 1978; Goldman, “Gating current kinetics in Myxicola giant axons,” Biophysical Journal, v59, pp 574-589, 1991; and L. Goldman, “Sodium Channel Inactivation from Closed States: Evidence for an Intrinsic Voltage Dependency,” Biophysical Journal, v69, pp 2369-2377, 1995, the entire contents of each of which are hereby incorporated by reference as if fully set forth herein). Of course, each state used to explain an observable behavioral effect may consist of one or more actual states.
In many circumstances, the temporal changes of probability for states for a particular protein are predicted based on an initial population distribution among the states, values for rate constants for the transition from one state to another, and a system of ordinary differential equations. Generally, such rate constants can be assumed constant only for a particular protein and fixed external forcing. Nevertheless, much progress is made in predicting cell processes under these conditions. The solutions require numerical computations that are effectively performed only using computers, iterative methods, and long iteration times with many time steps.
The process is used both to determine rate constants that match observations in the presence of various factors, e.g., to fit the rate constants to the observations, and also to predict the behavior of the biological system based on the rate constants determined independently of those observations, e.g., based on previously determined rate constants in earlier experiments in the same laboratory or published in the scientific literature.
Analytical solutions that do not require computationally intense iterative methods have been developed for fewer than four states and for four states when certain rate constants are eliminated (forced to zero). However, with no general analytical solution, many workers are forced to employ the computationally intense iterative methods over many small time steps to deduce the temporal changes in probability for a particular state or states. This impedes both the determination of rate constants to match data and the prediction of effects of an external factor on a system, given its effect on one or more rate constants.
Based on the foregoing, there is a clear need for an analytical solution for time changes in probability for a general four state biological system based on rate constants alone or in combination with initial population distributions of the four states.